Preheat the oven to 400 degrees F.
In a large oven-proof skillet or saucepan, heat the oil over medium heat. Add the garlic and cook, turning occasionally, until golden brown on all sides and very fragrant, about 3 minutes. Remove the cloves and discard.
Meanwhile, season the chicken generously with salt and sprinkle the cumin, coriander and allspice over the meat.
Raise the heat to medium-high and add the chicken in a single layer, spice-side down. Cook until nicely browned on one side, about 4 minutes. Remove to a plate.
Add the carrot and vinegar to the pan, scraping up any brown bits that may have formed on the bottom, followed immediately by the lime juice, honey, bay leaf, 1/2 cup water and 1/2 teaspoon salt. Bring to a simmer and reduce for 2 minutes.
Remove from the heat and return the chicken to the pan, seared-side up. Scatter the jalapeno over the top.
Transfer the pan to the oven and bake, uncovered, until chicken is fork tender, about 15 minutes. Let rest 10 minutes, then roughly chop or shred with a fork, top with mint, and serve alongside the tortillas, cabbage, mint or cilantro, radishes and lime wedges.